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Mirrors > Home > MPE Home > Th. List > rspsn | Structured version Visualization version GIF version |
Description: Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
rspsn.b | ⊢ 𝐵 = (Base‘𝑅) |
rspsn.k | ⊢ 𝐾 = (RSpan‘𝑅) |
rspsn.d | ⊢ ∥ = (∥r‘𝑅) |
Ref | Expression |
---|---|
rspsn | ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2828 | . . . . 5 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ (𝑎(.r‘𝑅)𝐺) = 𝑥) | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
3 | 2 | rexbidv 3297 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
4 | rlmlmod 19971 | . . . 4 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
5 | rlmsca2 19967 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
6 | baseid 16537 | . . . . . 6 ⊢ Base = Slot (Base‘ndx) | |
7 | rspsn.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
8 | 6, 7 | strfvi 16531 | . . . . 5 ⊢ 𝐵 = (Base‘( I ‘𝑅)) |
9 | rlmbas 19961 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
10 | 7, 9 | eqtri 2844 | . . . . 5 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
11 | rlmvsca 19968 | . . . . 5 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
12 | rspsn.k | . . . . . 6 ⊢ 𝐾 = (RSpan‘𝑅) | |
13 | rspval 19959 | . . . . . 6 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
14 | 12, 13 | eqtri 2844 | . . . . 5 ⊢ 𝐾 = (LSpan‘(ringLMod‘𝑅)) |
15 | 5, 8, 10, 11, 14 | lspsnel 19769 | . . . 4 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ ∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺))) |
16 | 4, 15 | sylan 582 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ ∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺))) |
17 | rspsn.d | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
18 | eqid 2821 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
19 | 7, 17, 18 | dvdsr2 19391 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (𝐺 ∥ 𝑥 ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
20 | 19 | adantl 484 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐺 ∥ 𝑥 ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
21 | 3, 16, 20 | 3bitr4d 313 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ 𝐺 ∥ 𝑥)) |
22 | 21 | abbi2dv 2950 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 {csn 4560 class class class wbr 5058 I cid 5453 ‘cfv 6349 (class class class)co 7150 ndxcnx 16474 Basecbs 16477 .rcmulr 16560 Ringcrg 19291 ∥rcdsr 19382 LModclmod 19628 LSpanclspn 19737 ringLModcrglmod 19935 RSpancrsp 19937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-mgp 19234 df-ur 19246 df-ring 19293 df-dvdsr 19385 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-sra 19938 df-rgmod 19939 df-rsp 19941 |
This theorem is referenced by: lidldvgen 20022 zndvds 20690 |
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